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Intuition is a priori knowledge or experiential belief characterized by its immediacy. Beyond this, the nature of intuition is debated. Roughly speaking, there are two main views. They are:
- Intuitions are a priori. This view holds that distinctions are to be made between various sorts of intuition, roughly corresponding to their subject matter (see George Bealer). The only intuitions that are relevant in analytic philosophy are 'rational' intuitions. These are intellectual seemings that something is necessarily the case. They are directed exclusively towards statements that make some kind of necessity claim. For example, a rational intuition is what occurs when it seems to us that a mathematical statement (e.g. 2+2=4) must be true. Intuitions as this view characterizes them are to be distinguished from beliefs, since we can hold beliefs which are not intuitive, or have intuitions for propositions that we know to be false.
- Intuitions are a species of belief, and based ultimately in experience. This view holds that intuitions are not especially different from beliefs, although they appear subjectively to be more unrevisable than other beliefs. Unlike the previous view, these intuitions are liable to differ between social groups. Evidence for this is shown in various psychological studies (e.g. the one by Stich, Weinburg and Nichols)
In the philosophy of Immanuel Kant, intuition is thought of as basic sensory information provided by the cognitive faculty of sensibility (equivalent to what might loosely be called perception). Kant held that our mind casts all of our external intuitions in the form of space, and all of our internal intuitions (memory, thought) in the form of time.
Intuitionism is a position advanced by Luitzen Egbertus Jan Brouwer in philosophy of mathematics derived from Kant's claim that all mathematical knowledge is knowledge of the pure forms of the intuition - that is, intuition that is not empirical (Prolegomena, p.7). Intuitionistic logic was devised by Arend Heyting to accommodate this position (and has been adopted by other forms of constructivism in general). It is characterized by rejecting the law of excluded middle: as a consequence it does not in general accept rules such as double negation elimination and the use of reductio ad absurdum to prove the existence of something.
In analytic philosophy 
In contemporary analytic philosophy, appeals to our intuitions are an important method for testing claims. A characteristic example is the post-Gettier literature concerning the analysis of knowledge. A philosopher proposes a definition of knowledge, such as the justified true belief account. Another philosopher constructs a hypothetical case where our inclination is to judge that the definition is met but the subject lacks knowledge or vice versa. Typically, this leads to the rejection of that account, though Brian Weatherson has noted that the weight placed on intuitions varies between different subfields.
Intuitions are customarily appealed to independently of any particular theory of how intuitions provide evidence for claims, and there are divergent accounts of what sort of mental state intuitions are, ranging from mere spontaneous judgment to a special presentation of a necessary truth. However, in recent years a number of philosophers, especially George Bealer have tried to defend appeals to intuition against Quinean doubts about conceptual analysis. A different challenge to appeals to intuition has recently come from experimental philosophers, who argue that appeals to intuition must be informed by the methods of social science.
See also 
- Immanuel Kant (1787) "Critique of Pure Reason", p35 et seq. http://www.gutenberg.org/catalog/world/readfile?fk_files=5144&pageno=35
- B. Weatherson, "What Good are Counterexamples?", Philosophical Studies, 115 (2003) pp. 1-31.
- M. Lynch "Trusting Intuitions", in P. Greenough and M. Lynch (ed) Truth and Realism, pp. 227-38.
- G. Bealer "Intuition and The Autonomy of Philosophy" in M. Depaul and W. Ramsey (eds) Rethinking Intuition: The Psychology of Intuition and Its Role In Philosophical Inquiry 1998, pp. 201-239.